We first blow up \(\underline{U}\times _{\mathbb {A}^1}\mathbb {A}^2\)  along the singular divisor \(D\times _{\mathbb {A}^1}0\)  to get \(\widetilde{\underline{U}(1)}\)  which is verified to be isomorphic to \(\mathcal {O}_{\mathbb {P}^1\times \mathbb {P}^1}(-1,-1)\) , and then contract one factor of \(\mathbb {P}^1\times \mathbb {P}^1\)  to obtain \(\underline{U}(1):=\mathcal {O}_{\mathbb {P}^1}(-1)^{\oplus 2}\) . This process gives the associated projection \(\underline{\pi }_{1}:\underline{U}(1)\rightarrow \mathbb {A}^2\) , which is called the expanded degeneration family of the standard local model \(\underline{\pi }:\underline{U}\rightarrow \mathbb {A}^1\) . One can refer to [[1]}, Section 1.1] for more details.
